US2643821A - Device for computation of roots of polynomials - Google Patents

Description

June 30, 1953 I... c. FRAGER 2,643,821

DEVICE FOR COMPUTATION OF ROOTS OF POLYNOMIALS Filed June 8, 1950 4 Sheets-Sheet l FIG. I

INVENTOR LOUIS C. FRAGER,

BY'W

ATTORNEYs June 30, 1953 c, A R 2,643,821

DEVICE FOR COMPUTATION OF ROOTS 0F POLYNOMIALS Filed June 8, 1950 4 Sheets-Sheet 2 FIG. 3

a, I [02 INVENTOR a0 Louis c. FRAGER,

BY WMMKQALMM ATTORNEYS June 30, 1953 L. c. FRAGER 2,643,821

DEVICE FOR COMPUTATION OF ROOTS OF POLYNOMIALS Filed June 8. 1950 4 Sheets-Sheet 3 FIG. 6

93 INVENTOR. LOUIS c. FRAGER ATTORNEYS L. C. FRAGER June 30, 1953 DEVICE FOR COMPUTATION OF ROOTS OF POLYNOMIALS Filed June 8, 1950 4 Sheets-Sheet 4 INVENTOR. LOUIS C. FRAGER ATTORNEYS Patented June 30, 1953 UNITED STATES PATENT OFFICE Application June 8, 1950, Serial No. 166,918 In France June 11, 1949 3 Claims; (01. 235- 61) wherein the generic term a denotes a real constant value, and wherein p is an unknown value p raised to the power q, the solutions of such an equation are n roots, which may have the ones or the others complex values or not. If a change of variable value is operated by assuming:

1) will be considered as a complex variable value of an argument or phase and of a modulus or absolute value p(7'=\/1). In the present specification the Words argument and phase are taken as synonymous and describe the angle which fixes the direction of a complex number. It results therefromthat Equation 1 becomes:

'IL v g pe- =0 3) q=0 If we consider an analytical function:

TL f(z )=z t q:

its roots will be given by the values of p and of 0 which render Hp) equal to zero. The function Hp) appears as a resultant of n vectors characterized by respective moduli a p and by respective arguments (10, i. e. the arguments of which form an arithmetical series.

If 0 is given the Value of 'rr/Z (or (2I C+1/2)7r), the end of the vector ,flp) will drawa particular locus in the complex plane of f. This will be the locus which is the image, in the plane of f, of a straight line 0=1"r/2 (or 0=(27c|-1/2)1r) situated in the plane of p.

This locus will be a boundary separating the region where the real portion of p is positive, from the region where this real portion is negative. V v

If successive curves f(-p) are drawn in the plane of f, for"successive constant values of 0, as many curves having constant values of 0 will be obtained, as one has chosen fixed values of 0; it is obvious that, by trying successive values, such values of 0 will be found for which the vector f(p) may be done equal to zero. It will suffice todraw as many loci having constant values of 0 as will be necessary to find those which pass through the'zero point. If for one of these solutions the value which is found for 0 is comprised between (2k1/2)1r and (2k;+1/2)1r, it will be concluded that there exists a root of f(p) having a positive real portion.

lf, one the contrary, the value found for 0 is comprised between (2lc+l/2)1r and (2k|3/2)1r, it will be concluded that there exists a root having a negative real portion. 7

In case if 0:0, there are'one or several positive real roots. v

In case if 0='1r, there are one or several negative real roots; I ,p

An object of the present invention is to provide cathode ray tube means for representing upon the screen of said tube taken as a complex variable plane, a straight line whose inclination with relation to areference line shows the phase of the complex variable and upon the same screen taken as a complex function plane, means to represent the curve representing the function of a variable having the said predetermined phase. M ,7

Another object of the present invention is to provide means for varying continually the phase of the complex variable; that is, the inclination of the straight line representing upon the screen of the cathode ray tube such variable until the curve representing the function upon the same screen passes through the point on the screen representing the origin of the complex function plane. a a

Another object of the present invention is to establish a correspondence between a plurality of successive points upon the straight line repre senting the complexvariable of a given phase and a plurality of successive points upon the curve representing the complex function.

Another object of the present invention is to provide meansfor determining the value of the modulus of a point of the straight line representing the complex variable "which corresponds to a point of the curve representing the complex function whichcoincides with the origin of the complex function plane, v

According toj a'iirst' feature of the invention the method of computation of roots of polynominals consists in the following operations:

To assume for the variable value of :0, a damped I alternative voltage, the amplitude of which e" (a being the attenuation factor or growith COIlv stant) represent the modulus of the complex variable value, and the phase of which, that is the product wt of the angular velocity by the time, represents the argument or phase of said variable value:

To assume for the real portion of the function Hp) the sum of n damped alternativevoltages in n circuits each representing one power of the variable value, and the resonant frequencies of which are integer multiple values of the fre-- quency of the A. C. voltage representing the variable value, and the logarithmic decrements of which are equal to each other, which circuits being charged at first respectively up to voltages proportional to the polynomial coefficients, and which are discharged all at a time; a

To assume. as an imaginary portion of the function Np) the sum of the above damped A. C; voltages, but the phases of which are each delayed To compose perpendicularly the real portion plus a voltage proportional to the polynomial constant term and the imaginary portion of f(p) in applying the representative voltages respectively to the pairs'of deflection plates of a cathode ray tube;

To stroboscope the spot of this C. R. T., by in creasing its brilliancy, at 27T/w intervals of time corresponding to arguments of p difiering from each other by the value of 211', so as to obtain, in the plane of a conformous transformedfigure, represented by a succession of luminous points, of a vectorial radius located in the plane of p, and having a constant argument 6;

To change the phase of the stroboscopic operation, so as to alter the argument of the variable value corresponding to visible points on the screen and so as to determine the value of this argument for which the pattern drawn by the luminous points passes through the zero point 'on the screen. This argument will be then'the argument of the root so disclosed, and the logarithmus of its modulus is proportional to the number of luminous points comprised between the'beginning of the discharge and the passage through the zero point.

According to a second feature, corresponding to an alternative embodiment of the invention. the method of computationof the roots, of a poly nomial consists in the following'operations:

To assume as a variable value 10 an aperiodic capacity discharge voltage the amplitude e of which (wherein a. is the reciprocal value of the time constant of the discharge circuit) represents the modulus of the complex variable value;

To assume for the real portion of the function Hp) the sum of n aperiodic voltages in n circuits representing each one a power of the variable value, and the time constants of which are integer dividers of the time constant of the circuit representing the variable value, and at first charged up to voltages respectively proportional to the polynomial coefficients, and which are discharged all at a time, each aperiodic voltage figuring in the sum being multipliedby cos no for the circuit of nth order;

To assume for the imaginary portion of the function flp) the sum of n aperiodic voltages in in circuits, pairwise identic to the former ones, each I aperiodic voltage figuring in the sum being multiplied by sin n0 for the nth order circuit;

To compose perpendicularly the real portion plus a voltage proportional to the polynomial constant term, and the imaginary portion of ,f(p), by applying the representative voltages respectively to the pairs of the deflection plates of a cathode ray tube;

To change'the values of 0 in the terms cos 116 and sin M, by suitably adjusting brushes of po-= tentiometers, and to determine the values of the argument 6 for which the curve drawn on the cathode ray tube screen passes through the zero point of this screen. The argument 0 is then the argument of the root so disclosed, and the logarithmus of its modulus is proportional to the time which is elapsed from the beginning of the discharge until the passage through the zero point.

The invention will be described hereunder in detail, with reference to three embodiments chosen in a nonrestrictive manner, the two first of which corresponding to the first feature of the invention, the third one corresponding to the second feature of the invention. These examples apply to polynomials of the third degree, but

they are obviously applicable also to polynomials of any degree.

This description will be made with reference to the accompanying drawings in which:

Figure l'represents the diagram of the electric resonant circuits used for generation of voltages which are'to be composed together, as well as the potentiometer enabling to adjust the initial voltages in condensers of said resonant circuits.

Figure 2 shows thediagram of a device for generation of isochronous pulses and a cathode ray tube. I

Figure 3 shows an alternative embodiment of the computor device according to Figure 1.

Figure 4. shows the diagram of a set of two electric circuits designed for an'aperiodic exponential discharge and which are used for generation of voltages which are to be partially composed, a potentiometer enabling to adjust the initial voltages in condensers of said discharge circuits, as well as potentiometers oi the respective circuits enabling to determine the fractions of voltages to be composed together.

Figures 5, 6 and 7 illustrate diagrammatically the mechanical devices for adjustment of th po tentiometer brushes in each circuit so as to collect the fractions sin 0, cos 0,. .sin n0, cos ne of the voltages generated in each circuit.

Figures 8, 9 and 10 represent illustrative diagrams of the curves appearing upon the screen of a cathode ray tube when the latter represents the plane'of the complex variable, then the plane of the complex function.

Fig. 11 represents the potentiometer allowing obtaining continuous voltages proportional to the coefilcients of the polynomial.

Referring now to Figs. 8 and 9, 02: represents the real axis in the complex variable plane and 0y the imaginary axis in said plane. In like manner OX represents the real axis in the complex function plane and CY the imaginary axis in said plane.

Let us assume that the polynomial to be resolved is whose roots are 2, 2|-2y' and -2-271 201, 202, 293, 204, 205, 205,- 281, 208 and 209 are vectors having arguments increasing from zero to 11' by stages of 1r/8 traced in the plane of the complex variable (Fig. 8).

own reference number. verging to the point of the abscissa ao=l6 which a series of otherpoints 222, 223

"between 223 and the origin to ale 5 1 the curvez l9.

upon'Fig. 8 are 2 and 0.

2 I, a 2 r2, 213, 2 It, 2 [5, 21 6, 2 l 1, 4 I18, 2 1'9 *are "curves traced intheplane of the complex function (Fig.9) and which are respectively the trans-- -*formed curves corresponding to the vectors having a reference number of 10 unitssmaller than their These curves are concorresponds to the origin of the' plane of' the com'pl'ex variable.

and that of the corresponding curve 2 I 5 is f a) reg-t 6p 3+16pg+ 6 r0 sp +lsi+j remp Let us consider upon the line 205 a point 22l and such that, the distance between22l andthe origino having a certain value an, the distance between'222 and the origin'should be equal to methe distance It results therefrom that the points 22 I, 222, 223 are at the intersection of the line 205 and an equiangular or logarithmic spiral "240 which makes (See Electromagnetiewaves by S. A. 'Schelkunoff, page 22, D. Van Nostrand).

There has been markeduponthe curve 215 the points 23! corresponding to the point 22L 232 corresponding to the point 222,' 233 corresponding to the point 223, etc. These points are represented upon Figs. 8 and 9 by small triangles.

There has also been marked and .shown by small circles containing a center the points 224, 225 and 22B upon the line "201 whose mutual distances obey the same law as the points of the line 205 and their correspondants 234,-235, 236 upon thecurve 2. The coordinates of the -point225 upon'Fig'B are 2 and 27'.

There has also been marked and shown by small triangles containing'a center the points221, 22B and 229-upon the line-2B9 whose mutual distances obey the same law 'as'the'p'oints of the line 205 and their correspondents231, 238; and 239 upon The coordinatesiof the point-229 Finally therehas been shown conventionally at the end of the line 20! an obliquelineandupon the "curve 2 (which-is no other than the'axis of .the abscissae of the:complex function plane) -a series of points represented byssmall oblique marks, 'at the end of theline'2fl2 a: small circle and upon the curve'2l2 aseries of points represented by small circles, at the: endof the line 203 a and upon the curv 2H6 ase'riesof points'represented by small crosses, finally at the 'end of line 208 a small square containing ai'point and upon the curve 2l8 a series of points represented by small squares each containing a point.

The curve 2l5 definesin the plane of Fig. 9 a

right hand region where the values of the real portion of the variable pare positive and a left hand region Where all the-values of the real porr tion'of thevariable p are negative. As the origin 0 of the plane of the 'com'plex functionis in the left hand region onecan deduce therefrom that the polynomial (4) has roots the realportion of which is negative.

The curve 2 I 1 passes through the origin 0 and has a point 235 which coincides withsuch origin. One deduces' therefrom that the point 225 is a point representative of a 'rootof the polynomial.

Thecurve 2l9 passes throughtheorigin O and has a point 239 which coincides Withsuch origin. One deduces therefrom that the point 229 is a point representative of a root of the polynomial.

In the solving of equations which isnow to be described, there will be made to'appear upon the screen of a cathode ray tube taken'successively as a complex variable plane and as a complex'function 'plane on one hand afamily of lines such as the lines 2!!! to ZIlQand upon the other hand corresponding transformed curves2ll to 2m. The center of 'thescreen of thecatho'de'ray tube being at the same time the origino of the coordinates of the plane of the variable and the origin 0 of the coordinates of the plane of thefunction'will be designated by the reference character "00. Finally, according to the-case, the lines of Fig. 8

and the curves of Fig. 9 will'beinaterialized upon the screen of the cathoderay tube either by a luminous discontinuous line of pointsor by a' continuous luminous line.

Finally, it will'be noted that the curves of Fig. 9 converge to the abscissa pointao. Inorder to know the sign'of the real portion of the roots, it amounts to the same'thing to knowonwhatside the origin 0 of the coordinates" is located'with relation to the curve 215 passing through the abscissa point (1001' to 'know'on which side the 'vided to be able to place the converging point'of the curves 2H to H9 either at the'abscissa" point an or at the origin of the coordinates of the complex function plane. The translationof the difierent curves will be *madeyaswillbeexplained below, by means of a switch I02.

Referring toFig. l, I00,200;and 300 designate generally three oscillating'circuits, each comprising a condenser, a first inductance and a resistance in parallel with a second inductance. The circuit I is composed of'the' condenser II, the resistance l2 in par'allel'with the inductance l3 and the inductance l4. The circuit 209 comprises the condenser 2!, the resistance 22' in parallel with the inductance 23 and the inductance 24. The circuit 300 comprisesthe'condenser' 3|, the resistance 32 in parallel-with the inductance 33 and the inductance 34.

Each of the circuits comprises a three-way switch having the contacts H3, H1" and Ill for the circuit I69, 20, 28 and 20" for the circuit 200, and SU, 30" and'3il" for the circuit300. The median positionof eachoswitchisan idleposition, the left hand position [0,20' and-30' is'an operative position :used for charging thecondenser of the'respective circuits. and the right hand position ill", 20"and30" is antoperative position used for the oscillating discharge of the volts. The positive terminal 4 I of the potentiometer 5 is permanently connected to theiplate l8'of the condenser l l-by the connection.

Upon the potentiometer there can slide with overlapping four sliders l, 2, 3, and 4. Upon the potentiometer "35 there can slide with overlapping three sliders c2, 43, and 44. The difference in potential between the terminal ll and the slider i represents the coefficient c1 of the polynomial. The difference in potential between the terminal El and the sliders 2,3, and 4 represent respectively the coefficients a2, as and do when the latter have the same sign as on. The difference in potential between the terminal E9 and the sliders 42, 33, and 6G represent respectively the coefiicients a2, a3, and do when the latter have a sign opposite to m.

When the switch ID is in position 10', the plate is of the condenser I I is brought to the potential of the slider l by the slide 5! of such slider and the connections 57 and 55. The condenser II is charged with a voltage proportional to m (it is assumed that 121 .is positive which is always permitted in multiplying if it is necessary the first member of the polynomial (4) by l).

When the switch 29 is in position 2% and as the coeificient a2 is positive, the plate 23 oi the condenser ilfis brought to the potential of the positive terminal at by the connections 58 and 58, the movable contact {ii of the switch 1-8 in its upper position, and the connection 66. The plate 29 is brought to the potential of the slider 2 by the slide 52 of the slider, the connection F, the movable contact 86 of the switch 58 in its upper position and the connection 61.

When the switch Ell is in position iii and. as the coefficient as is negative, the plate 28 of the condenser 2| is brought to the potential of the negative terminal 4!] of. the battery through the connections 88 and E9, the movable contact 6| oi the switch 18 in its lower position and the coir nection G6. The plate 2% is brought to the potential of the slider t2 by the slide $2 of such slider, the connection IBI, the movable contact 66 of the switch 58 in its lower position and the connection 67.

It is the same when the switch 38 is in position 30' and as the coefficient as is positive, the plate 38 of the condenser 3| is brought to the potential of the positive terminal 41 by the connectionsfit and 72, the movable contact l! of the switch being in. its upper position and the connection 75!. The plate 3% is brought to the potential of the slider 3 by the slide 53 of such slider, the connection 15, the movable contact it of the switch :19 being in its upper position and the connection Hi.

When the switch 30 is in position 35 and as the coefficient a3 is negative the plate 38 of the condenser 3! is brought to the potential of the negative terminal iil by the connections 58 and i9, the movable contact ll of the switch 39 being in its lower position and the connection 70. The plate 39 is brought to the potential of the slider 3 by the slide 63 of such slider, the connection 73, the movable contact P5 of the switch 9 being in its lower position and the connection Ti.

Finally, according to whether the switch I is in its upper or lower position, the movable contact I l of such switch is connected either by the connection 85 to the slide 54 of the: slider 4 or by the connection 84 to the slide 64 of the slider M and the movable contact 18 is connectedeither by the connections 58 and 82 to the positive terminal 4| or by the connections 68 and 86 to the negative terminal of the battery. One disposes therefore between the movable contacts 14 and 18 of the switch I50 a potential difference either equal to that of the the terminals 4! and 92 to which are soldered the ends of the wire 99 and to which are connected the poles of t e battery I.

The mounts 9% and 8! are connected by a metallic rule 93 which forms the base of the potentiometer and which is graduated upon its upper portion into 100 divisions. Four conducting slides, of'which only three are shown, 5!, 52, and 53 are fixed to the two mounts and terminate upon mount 9! in the terminals M, 95, 96 and 97 to which are fixed respectively the connections El, 65, '15 and 35. Upon the slides 5!, 52, 53 and 54 (the latter not shown in Fig. 11) the sliders l, 2, 3 and :3 can be respectively displaced (the latter is not shown in Fig, 11) The sliders have pointers attached thereto such as 98 which are displaced in front of the graduated rule 93 which allows positioning them opposite a given. graduation,

In order to position the sliders one considers what is (or what are) in the polynomial (4) the highest coefficient in absolute value. One finds:

The sliders I and 3 are placed upon the graduation 100, the slider 2 upon the graduation If all the coefficients are positive the switches 48, 49 and 15!] are in their upper position. If certain coefiicients are negative, for example as 1 a2: 6

' impedance with relation to the resistance l2 (or 22, or 32) and only intervenes in order to obtain at its terminals a voltage in phase with the discharge current, while the voltage at the terminals of M is in quadrature with the discharge current. The assembly of a resistance, such as l2, in parallel with an inductance, such as I3, can be considered as having an impedance practically equal to the resistance I2.

Assuming therefore in the circuit 180, R1 as the value or" resistance [2, L1 the value of inductance 14, C1 the capacity of the condenser II, we have:

aceasai where? art. is the attenuation. coefficient of. the=- discharge current andwi its angular rvelocity;

It is the same in circuits 200 and 300:

it. 2L2 as 2L3f 1 2 not? L L 63 Where R2, L2, Lo; 02, C-are respectively the' values of the elements 22, 32, 24, 34; 2|, SI; (12, 113

the attenuation factors .and wz, m3 the angular velocities of the circuits; 200 and 300.

It is arranged by construction in order that that is; in order that in all the circuits the quotient If, therefore after having; thrown. thex switches. IE; 2.0,; and 130- towardsthe left in order to charge thexcondensers I I, 2 I, and .3I .undervoltages respectively proportional to the algebraic values $1,112, and as, one: throws them towards. the. right, the-circuits I00, 200, and 309: arethe seats-of oscillating discharges; and. the discharge currents i imthe threeci-rcuits, will-.be, with the exception o-fvacoeflicientconstant. OfzDIOIJOItiOIl, equal to:

iZZHZeT 003* 2101'" is'=.asecos 3wt- The winding I4 of the circuit Hill-"is coupled-to a first secondary Winding; It; the. winding 2410f the-circuit 200 *to a firstr secondary winding 26' andthe winding:34 of-Jthe circuit 3illiitoafirst secondarywinding- 36. The transformation ratios.-

of the transformers constituting the windings I4" and I6, M ami-26.; 3fl-and 3 62 are. the same. The secondary windings I6, 25'; audits; are. placedin series and the voltages'developed at their termina-lsare-applied additively by means of the. connections 89 l and 83-; tozthe-horizontal deflection plates =46 of-a cathode rayvtuloe" I01. The'voltage at the terminalsof the primary windings It, 26; and 34 is as indicatedinquadrature. with thedischarge current. The voltage developed at the terminals of each of thesecondary windings I 6,- 26; and- 36 is therefore also inquadrature with the dischargecurrent of the corresponding cirouit and i the voltage 1 applied :loietween; the: plates this ,under theseconditions equal withv the exception cfra, proportion factor; to

that is. to the imaginary portion of flp) The Winding I4 of the circuit IGiLis coupled toga secondsecondary winding I5, the winding 24 of the circuit 200 to a second secondarywinding 25,.andthe winding. 34 of the circuit 300 to a.second secondary winding 35. The winding I3 of the circuit I 00 is coupled to a secondary.

winding II, the winding23 of the circuit 290 to a secondary winding. 21 and the. winding 33 of thecircuit 306, to, a I secondary winding 31. The secondary windings I1, I5,. 21, 25, 31, 35. are

placedin series. andthe voltages developed at their, terminals are appliedadditively with the voltage existing between the. movable contacts "and. 7B which is proportional .to the algebraic.

value of a0, across the connections 8|, 81, and BE to the vertical deflection plates 41 ofthe. cathode ray tubeIOI.

as, indicated almost in phase with the corresponding discharge current. The voltage at lthe, terminals ofeach of :the secondary windings I7, 21 and 3! is consequently almost in phase opposition with the corresponding discharge current. The voltage at the terminals of each of the primary windings I4, 24, and 3 liislin quadrature with the corresponding dischargecurrent. voltage-at the. terminals of each of the secondary windings I 5, 25, and 35 is consequently also. in quadrature withthe corresponding discharge current. The secondarywindings, I5, 25..and 35 have only a small number of wire turns. so that the I voltagei111 quadrature that they introduce compensates the voltage inuquadratureintro.- duced by thegsecondary windings I1, 21 and 3'1. and two transformers, such as I5 and. IT, in-. troduce a resultant voltage. exactly in phase with thehcorresponding discharge? current. The voltage; appliedbetween the plates ,4! is under these. conditions equal, with the exception of. a. proportion factor, to:

cos 2wt+a1' cos ot-I-ao thatis, to the. real portion of flp).

In the preceding explanations the, switch I02 which connects the. wires Mend 87 issupposed to be open. If this switchtisclosedone cancels in the above-expression (8) the term..ao.

If inplaceof throwing the three switches "1,20,

and 30 towards the right one should only throw.

the switch I!) .towards the right and if the switch I02 is vclosed which: eliminates the placingv in series with the plates 41 of-a voltage proportional to .ao-the, voltages applied ,to the pairs of plates. 41 and 46 will be respectively:

a e OOSwt I a e S111 wt The spot describes upon the screen a logarithmic spiral whose first expression (9) represents the abscissa and the second expression the ordinate and which makes with its radius victor an angle t thus that (see Schelkunofi cited) This logarithmic spiral is thespiral; 240- of Thevoltage at theterminals of eachof the-primarywindings I3, 23, and 33. is-

The a the three-way switch 509.

Fig. 2 represents a circuit designed to illuminate periodically the spot of the cathode ray tube llll. This circuit may produce impulses of positive polarity at recurrent frequency and with variable phase. Such circuits are well known in the art and that of Fig. 2 is only shown for illustrative purposes.

IDI is the cathode ray tube shown in Fig. 1, I04 is the cathode heated by the battery IZl, I03 is the grid, I05 the screen, 55 the horizontal deflection plates and i! the vertical deflection plates.

The potentiometer I06 is fed by the battery IIB'I through the connections Hi3 and II! and The potentiometer I06 comprises two sliders H2 and H3. The slide lid of the slider H2 is connected by the connection H5 to one of the plates or a condenser H5 whose other plate is connected by the connection Hi to the negative terminal I08 of the battery.

The condenser H6 is shunted by a neon tube I I8 and the primary winding H9 of a transformer. The secondary winding IZI of such transformer is connected at one terminal to the point I20 at the junction of the neon tube H8 and the primary winding H9 and, by its other terminal, to the grid I93 by the connection 522. The switch me has one median idle position and two operative positions, the one Hi9 where the connection I It is connected to the potentiometer Hi6 and the other I09" where the connection IIII is connected to a slide I23 of the slider H3.

When switch I09 passes from the median position to the upper position I99, the condenser I I6 3.

the switch is in position I09". The condenser I It is then charged with a voltage which is assumed to be smaller than the disruptive voltage of the tube H8. If one sharply swings the switch I09 from its position I99 to its position I09 condenser lit, already charged under a voltage equal to will be further charged under a supplemental voltage equal to number of impulses applied to the grid W3 start is charged exponentially starting from a zero 7 value to the potential difference existing between the slider H2 and the point Hi8. When the voltage at the terminals of H6 reaches the value of the disruptive voltage of the neon tube M8, the

condenser discharges into such tube. There results therefrom a voltage surge at the terminals ofthe primary H9 and at the terminals of the secondary IZI and the sum of these voltage surges is applied by the connection I22 to the J grid I83. This grid is normally at the potential of the point I08. During the appearance of the voltage surge at the terminals of I I9 and I2! it is carried to a positive potential with relation to the point I33 and the electronic beam of the tube IBI is unblocked for a short instant. The repetition period of the positive impulses applied to the grid I03 may be varied by displacing the slider IIZ upon the potentiometer I65.

Let us assume that the battery It? has an electromotive force E.

Let us designate by d the distance between the end I25 of the potentiometer I05 connected to the negative terminal of the battery It? and the point I25 connected to the positive terminal of the battery It? when the switch I89 is in position I69. Let us designate by x the distance between the point I2 i and the slider H2. When the switch I99 passes from its median inopera tive position to the position 109 the condenser I I6 is discharged with a voltage and this charge stops before reaching such value when the increasing voltage at the terminals of I It becomes equal to the disruptive voltage of the tube I I8.

Let us designate by y the distance between the point I25 and the slider H3 and let us suppose ing from the swinging of the switch I69.

The operation of the apparatus is therefore the following:

One. positions the slider I, and those of the sliders 2, 3 and t which correspond to the positive coemcients of the polynomial and those Of the sliders 42, 43, and M which correspond to the negative coefficients, while taking care to associate with a positive coefficient a slider of the potentiometer 5 and a switch d8, 49, or I55 in the upper position and with a negative coefficient a slider of the potentiometer 55 and a switch 33, 13 or I50 in a lower position.

This being done, and the switch Hi2 being in closed position, one swings the switch iii towards the left into the position Iii tocharge the condenser II under a voltage proportional to m and one swings simultaneously the switches I0 and Itifiinto the respective positions Ill" and I99. The spot if it was constantly illuminated would describe a logarithmic spiral such as 2 56 of Fig. 8 or 340 of Fig. 1'0, but in fact as it is illuminated .only at regular intervals one observes a series of points 3M, 362, 383, 30 3, (Fig. 10) located in any way whatsoever upon the spiral 346. The preceding operations are recommenced several times while modifying at each time the position of the slider I I2 until the points observed become aligned with the center 00 of the screen I65 such as shown at 3M, 31D, 32B, 330.

The repetition period of the periodic illuminations of the screen being determined by a suitable positioning of the slider I12 there is carried out successively the following pairs of operations, for different positions of the slider H3:

First operation: the switches I0 and I09 are swung into the positions I0 and I99" to charge the condenser II and to adjust to a given value the phase of the illuminations of the spot. Then, the switch I62 being closed one swings said switches into the positions ill" and H39 and one observes upon the screen an alignment of points upon a given line of inclination such as the lines 26! to 2GB of Fig. 8 or L352 of Fig. 10. A circular degree scale 304 is provided upon the circumference of the screen I65 (Fig. 10) in order to measure the inclination of said alignment.

Second operation: the switches II], 25!, 30, and

ductance 535, 555 is connected by the coil 555. and the switch 535to the connecting wire '81.

The connecting wire 8!! is connected to the grid 55!; of the tube 55!.

The switch i5 is a three position switch, the median position being inoperative, the left hand. position connecting the output of the coil 5H5 directly to the grid 554 through the connection 55! and the right hand position connecting V the output of the coil 5H5 to the point 525.

The coils 5H6, 525, 536 have a number of turns which are respectively half those of the windings 5M, M5; 523, 525F and 533, 53 i and are con pled without leakage to the latter.

One of the horizontal deflection plates 45 of the cathode ray tube NH is connected by the connection 55 to one end of the chain of secondary windings EN, 521 and 531, respectively coupled vith the primaries SIS, 5!4;'523, 524 and 533, 532- and the other end of such chain is connected by the connection 83 to the other horizontal defiection plate 45.

The cathode 552 of the tube 55! is polarized by the resistance 555 and the plate 555 is fedby the battery 55?. Moreover the anode circuit of 55! comprises a primary winding 558 which is coupled to a secondary winding 555 connected by the connections 8!, 555, and 83 to the vertical deflection plates 5'! of the cathode ray tube A switch I62 connected between the wires 85 and 8'! prevents when closed the introduction in series upon the chain going from the cathode 552 to the grid 555 of the voltage proportional'to the a coefficient an.

The voltage at the terminals of the assembly formed by the primary M3 and the secondary 5l6 is in phase with the discharge current and equal to the product of said discharge current by the resistance of the windings 515 and 5H5. The self-inductance of 5 3 being the mutual inductance between 5M, and 5H; is, 7

assuming the equality of the number of turns of 5m and SIS and their coupling without leakage, also equal to The voltage drop (see Equation 11) above, at the terminals of the assemblies 523, 526 and 533, 555

voltages in phase with the current.

If therefore the switches 5I5, 525, and 535 are closed (towards the right as regards switch 5i5) the voltage applied between the cathode and the grid of the tube 55! is proportional to the real portion of flp) given by the Equation 8. The

voltage at the terminals of the deflection plates ii is also proportional, with the exception of the sign, to such real portion. I

The voltage at the terminals of the secondaries 5!|', 521, and 53'! is in quadrature with the discharge current in the corresponding circuit. The voltage at the terminals of the deflection plates 45 is proportional to the imaginary portion of f0 given by the Equation '7.

The operation for solving an equation by the device of Fig. 3 is almost similar to that for solving an equation by the construction of Fig. l.

The switches cm, 520 and 535 being swung towards the left and the switches 5l5, 525 and 535 being open, the condensers are charged with voltages proportional to the coefiicients of the polynoinial.

If one swings the switch 5H3 towards the right and the switch 5l5 towards the left there is obtained upon the screen 15 a series of points marking out a line such as 20! to 209 of Fig. 8.

If one swings the switches 5!@, 525 and 530 and the switch 5! 5 towards the right the switches 525 and 535 being closed and I92 being open, there is obtained upon the screen a series of points marking out a curve such as 2!! to 2!S of Fig. 9. 1

' The search for the sign of the real portion of the roots and the value of the argument of the roots is made by displacing the slider H3 as has been indicated. The determination of the value or" the modulus is made in a way exactly similar to that set forth precedingly and it is not necessary to repeat the'explanations already made.

In the two applications which have just been described one has taken for the argument 0 of the variable p the quality wt variable as a function of the time. Whence it is necessary in order to obtain the corresponding transformation in the plane of ,f, of a radius vector in the plane of p having a given 0, having as its period to introduce a recurrent st'roboscopic effect by means or" synchronized scintillations.

In the third application which is now going to be described with relation to Fig. i the vector representing the variable 21 is no longer a rotating vector and the discharge circuits are no longer oscillating but aperiodic. 7

Upon Fig. 4, 45!, 502, 493, 45!, 552 and 453. are siX similar aperiodic circuits (4!)! being identical with 45!, 452 with 452 and 483 with 453). They each comprise a condenser 4H, 42!, 43!, 55!, 47! and 48! respectively and a potentiometer resistance H2, 422, 432, 462, 412, and 482 respectively. The time constant of the circuits 402 and 452 is two times smallerthan the time constant of the circuits 4!!! and 55! andthat of the circuits 453 and 553 is three times smaller. This result canbe obtained either by giving to the condensers 42! and 41! a value two times smaller than to thecondensers 4!! and 45! and to the condensers 43! and 48! a value three times smaller, the resistances being equal in all the circuits, or by giving to the resistances 422 and 4'52 a value two times smaller than to the resistances M2 and 452 and to the resistances 532 and 582 a value three times smaller, the condensers being equal in all the circuits.

Each circuit comprises a three position switch, the median position being inoperative, the left hand position corresponding to the charging of the condenser in the circuit and the right hand 17 position to the discharging of the condenser. These switches are M9 and 469 for the circuits 40! and 45!, 426 and 476 for the circuits 402 and 452, 436 and 489 for the circuits 403 and 453. The left hand positions are indicated at M9, 469', 423', 470, 439' and 466 and the right hand positions at M9", 469", 426", 416", 436" and 48D".

The system of potentiometers and of sliders for obtaining voltages proportional to the coefficients of the polynomial is the same as in the case of Fig. l and is not shown. It is schematized in the form of four pairs of leads 59, 31; 69, 31; l0, l1; and 89, 81, such that between the two leads of a pair there is a voltage proportional to the algebraic value of a coefficient. The value of the coeificient is written opposite a bracket connecting the two leads of each pair.

The wire 59 is connected to the plates M6 and 466 of the condensers 4H and 46!, the wire 69 to the plates 428 and 478 of the condensers 42! and 4', the wire 19 to the plates 438 and 463 of the condensers 43! and 481.

The wire 57 is connected to the left hand contacts of the switches 4| and 460, the wire 6? to the left hand contacts of the switches 429 and 476, the wire H to the left hand contacts of the switches 439 and 489.

Upon the potentiometers M2, 422, 432, 462, 472 and 482, are displaced the sliders M3, 423, 433, 463, 413, and 483 respectively and their respective displacements are combined in such a way as to pick up (0 being a predetermined angle):

Upon the circuit 41H cos 0 Upon the circuit 492 a cos 20 Upon the circuit463 a cos Upon the circuit 45! a sin 0 Upon the circuit 452 a sin 26 Upon the circuit 453 a sin 30 a fraction proportional to fraction proportional to fraction proportional to fraction proportional to fraction proportional. to

fraction proportional to 425 and the slider 433 to the connection 61 by means of a switch 435. The connection 66 is connected to the other vertical deflection plate 4'1.

On of the horizontal deflection plates 46 of the cathode ray tube i0! is connected to the point 464 of the circuit 45l through the connection 89, the slider 463 to the point 424 of the circuit 452 by means of a switch 465, the slider 413 to the point 484 of the circuit 453 by means of a switch 415 and the slider 483 to the other horizontal defiection plate 46 by means of a switch 465 and a connection 63.

The switches M5 and 465 are three position switches, the median position being inoperative, the left hand on is that in which a connection is established between the slider M3 or 463 and point 424 or 414 and the right hand one that in which a connection is established between th slider 413 and the connection 86 and between the slider 463 and the connection 83.

The switch I92 when in the closed position as where a, designates the inverse of the time constant of the circuits 40! and Figs. 5, 6 and 7 show the system for placing into place simultaneously the sliders upon the potentiometers. A knob 493 fixed upon a shaft 490 has a pointer 492 which is movable in front of a dial 49I graduated in angles 0. Upon Fig. 5 for example the pointer i opposite the graduation 0:20.

Upon the shaft 496 (Fig. 6) are mounted three gears 469, 479 and 489 which mesh respectively with the gears 419, 429, and 439. The gearing ratio between the gears 469 and 489 is equal to unity, the gearing ratio between the gears 4'59 and 429 is twice that and the gearing ratio between the gears 489 and 439 is triple that.

The gears M9, 429 and 439 carry crank pins 4H, 42?, and 431 respectively which actuate respectively and by pairs the connecting rods M6 and 466, 426 and 416, 436 and 486.

The connecting rod 4? drives the slider 413 which slides on the guide 404 and the connecting rod 466 drives the slider 463 which slides on the guide 454 (Fig. 7). The length of the connecting rods is suificiently large in order that their inclination with relation to the guides may be negligible. The sliders M3 and 463 slide along th potentiometers M2 and 462 which are linear and are located at right angles to one another. Under thes conditions there is received upon the sliders M3 and 433 voltages respectively proportional to the fractions cos 0 and sin 0 of the voltage at the terminals of the potentiometers 412 and 462.

The same assemblage which has just been described for the circuits 49l and 45! and as shown in Fig. '7 is. similarly provided for the circuits 462 and 452 on one hand and 493 and 453 on the other hand. The only difference is the following:

When the knob 493 is displaced so that the pointer 492 leaves the zero graduation and is placed opposite to the mark 0, the crank pin 4!! rotates for an angle 0 but the crank pin 42? rotates for an angle 20 and the crank pin 431 rotates for an angle 36 because of the gearing ratios of the gears 419, 429 on one hand and 489, 439 on the other hand. The voltages re ceived will therefore be proportional to cos upon the slider 423, sin 20 upon the slider 4T6, cos 36 upon the slider 433. and sin upon the slider 483.

it is clear under these conditions that during the discharge of the circuits, the voltage applied between the plates 47 is no other than the real portion of f(p) given by the Equation 8 (switch I92 open) and that the voltage applied between the plates 46 is no other than the imaginary portion of f(p) given by the Equation '2.

The operation of the device of Fig. 4 is then as follows:

In order to charge the condensers of the circuits the switches 410, 429, 430, 469, 410 and 439 are swung into the positions 4H1, 420', 439, 469, 416 and 489' with the switches M5, 425, 435,

465, 415 and 485 in open position.- Then the knob 493 is turned to place the pointer 492 opposite a given graduation 0. Then the switches M5 and 465 and the switches 410 and 460 are swung towards the right. There is then observed upon the screen I05 of the cathode ray tube l! one of the lines 2M to 209 of Fig. 8 according to the value of 0. But'instead of the lines being marked ofi by a discrete series of luminous points, they appear as a continuous line. If

for example, one will observe upon the screen the line 205.

Then the condensers 4| I and 46! are recharged and simultaneously the switches 4E5, 425, 435,.

565, 415, and 485 are closed (towards the left as regards the switches M5 and 465) and at the same time one swings towards the right the switches of all the circuits. One may note that all the switches 415, etc. which may betermed adding switches may remain open when the switches 4H0, etc. which may be termed circuit switches are in charging position and may be closed when the circuit switches are in discharging position. One observes then upon the screen H35 of the cathode ray tube lfll one of the curves 2H to ZIS of Fig.9, for example the curve 2l5 if one has taken as assumed But instead of the curves being marked on by a discrete series of luminous points they appear as a continuous line. The center 00 of the screen I05 being at the left of the curve 2 I 5 one deduces therefrom that the polynominal has roots with a real portion negative.

One recommences the preceding operations during the course of which the screen IE5 appears successively as the plane of the complex variable 17 and as the plane of the complex function f, for different values of 0 until a curve is obtained passing through the center 00 of the screen. The corresponding angle 0 is then the argument of a root.

In order to secure the modulus of such root, a *chronometer is started at the beginning of the discharge, that is at the moment when all the adding switches being closed and the switch I02 open one swings all the discharging switches towards the right and one stops such chronometer at the moment when the spot describing the curve passes through the center 00 of the screen. Let ts be the time registered. One has: s e-s from which p8 by logarithmic calculation.

Although the invention has been described with specific examples of application it is possible to make therein various variations which are easily developed by a worker in this art and which can be considered as coming within the scope of the invention.

What I claim is:

l. A computing device for determination of the roots of polynomials and of the sign of the real portion of the same comprising in combination a first circuit corresponding to the variable of the polynomial to be solved and comprising at least a first resistor and a first capacitor, a second circuit corresponding to the square of the variable and comprising'at least a second resistor having a value equal to half the value of the first resistor and a second capacitor, a nth circuit corresponding to the nth power of the variable and comprising at least a nth resistor having a value equal to the nth fraction of the first resistor, and a nth capacitor, whereby all the circuits have attenuation factors proportional to the terms of an arithmetic series of a ratio 1,

commutation means for respectively charging the capacitors of these circuits under voltages proportional to the constant coeificients of the first, second, nth powers of the variable in the polynomial to be solved, commutation means for simultaneously discharging the capacitors of said circuits, means for picking-up a first fraction of the discharge current of the first circuit equal to the cosine of a given angle and a second fraction of said discharge current equal to the sine of the said angle, means for picking-up a first fraction of the discharge current of the second circuit equal to the cosine of two times the given angle and a second fraction of said discharge current of the second circuit equal to the sine of two times the given angle, means for picking-up a first fraction of the discharge current of the nth circuit equal to the cosine of n times the given angle and a second fraction of said discharge current of the nth circuit equal to the sine of n times the given angle, means for obtaining a first sum of voltages comprising the voltage fractions equal to the cosine of the given angle and to the cosines of its integer multiples and a D. C. voltage proportional to the constant coefficient of the polynomial, means for obtaining a second sum of voltages comprising the voltage fractions equal to the sine of the given angle and to the sines of its integer multiples, a cathode ray tube, means for applying the first sum of voltages to the first pair of deflecting plates and the second sum of voltages to the second pair of deflecting plates of said cathode ray tube, means for varying the given angle until the curve drawn by the spot on the tube screen passes over the center of the screen and means for measuring the time interval between the beginning of the discharge and the passage over the center of the screen, whereby the phase of one root of the polynomial is equal to the angle for which the curve passes over the center of the screen and the modulus of said root is proportional to the logarithm of said interval.

2. A computing device for determination of the roots of polynomials and of the sign of the real portion of the same comprising in combination a first aperiodic circuit corresponding to the variable of the polynomial to be solved and comprising a first resistor and a first capacitor, a second aperiodic circuit corresponding to the square of the variable and comprising a second resistor having a value equal to half the value of the first resistor and a second capacitor, a nth aperiodic circuit corresponding to the nth power of the variable and comprising a nth resistor having a value equal to the nth fraction of the first resistor, and a nth capacitor, whereby all the circuits have time constants proportional to the terms of an arithmetic series of ratio 1, commutation means for respectively charging the capacitors of these circuits under voltages proportional to the constant coefficients of the first, second, nth powers of the variable in the polynomial to be solved, commutation means for simultaneously discharging the capacitors of said circuits, potentiometric means for pickingup a first fraction of the discharge current of the first circuit equal to the cosine of a given angle and a second fraction of said discharge current equal to the sine of the said angle, po-

tentiometric means for picking-up a first fraction of the discharge current of the second circuit equal to the cosine of two times the given angle and a second fraction of said discharge current of the second circuit equal to the sine of two times the given angle, potentiometric means for picking-up a first fraction of the discharge current of the nth circuit equal to the cosine of n times the given angle and a second fraction of said discharge current of the nth circuit equal to the sine of n times the given angle, means for obtaining a first sum of voltages comprising the voltage fractions equal to the cosine of the given angle and to the cosines of its integer multiples and a D. C. voltage proportional to the constant coefiicient of the polynomial, means for obtaining a. second sum of voltages comprising the voltage fractions equal to the sine of the given angle and to the sines of its integer multiples, a

cathode ray tube, means for applying the first sum of voltages to the first pair of deflecting plates and the second sum of voltages to the second pair of deflecting plates of said cathode ray tube, potentiometric means for varying the given angle until the curve drawn by the spot on the tube screen passes over the center of the screen and means for measuring the time interval between the beginning of the discharge and the passage over the center of the screen, whereby the phase of one root of the polynomial is equal to the angle for which the curve passes over the center of the screen and the modulus of said root is proportional to the logarithm of said time interval.

3. A computing device for determination of the roots of polynomials and of the sign of the real portion of the same comprising in combination a first oscillatory circuit coresponding to the variable of the polynomial to be solved and comprising a first resistor, a first capacitor and a first inductance, a second oscillatory circuit corresponding to the square of the variable and comprising a second resistor having a value equal to half the value of the first resistor, a second capacitor and a second inductance having a value equal to the fourth of the value of the first inductance, a nth oscillatory circuit corresponding to the nth power of the variable and comprising a nth resistor having a value equal to the nth fraction of the first resistor, a nth capacitor and a nth inductance having a value equal to the square of the nth fraction of the first inductance, whereby all the 22 simultaneously discharging the capacitors of said circuits, transformer means for respectively picking-up voltages proportional to and cophasal with the discharge currents of the circuits, transformer means for respectively picking-up voltages proportional to the discharge currents of the circuits and having a phase-shift of with respect to the latter, means for ob taining a first sum of voltages comprising the voltages proportional to and cophasal with the discharge currents and a D. C. voltage proportional to the constant coeflicient of the polynomial, means for obtaining a second sum of voltages comprising the voltages proportional to the discharge currents and having a phaseshift of 90 with respect to the latter, a cathode ray tube, means for applying the first sum of voltages to the first pair of deflecting plates and the second sum of voltages to the second pair of deflecting plates of said cathode ray tube, pulse generator means for periodically illuminating the spot of said cathode ray tube, said pulse generator having a recurrence frequency equal to the frequency of the first oscillatory circuit, means for varying the phase of the pulse generator with respect to the beginning of the simultaneous discharge until the curve drawn by the periodically illuminated spot upon the cathode ray tube screen passes over the center of the screen and means for measuring the time interval between the beginning of the discharge and the passage over the center of the screen, whereby the phase of one root of the polynomial is equal to the phase of the pulse generator with respect to the beginning of the dicharge for which the curve passes over the center of the screen and the modulus of said root is proportional to the logarithm of said time interval.

LOUIS C. FRAGER.

References Cited in the file of this patent UNITED STATES PATENTS Number Name Date 2,324,851 Koch July 20, 1943 2,349,437 Keeler May 23, 1944 2,454,549 Brown Nov. 23, 1948 FOREIGN PATENTS Number Country Date 607,397 Great Britain Aug. 30, 1948 OTHER REFERENCES An Electromechanical Method for Solving Equations, Schooley; RCA Review, volume III, No. 1, July 1938; pages 86-96.

An Electrical Algebraic Equation Solver, Herr and Graham; Review of Scientific Instruments, volume 9, October 1938; pages 310-315, inclusive.

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